DisC Diversity: Result Diversification based on Dissimilarity and Coverage
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Transcript of DisC Diversity: Result Diversification based on Dissimilarity and Coverage
DisC Diversity: Result Diversification based on Dissimilarity and Coverage
Marina Drosou, Evaggelia PitouraComputer Science DepartmentUniversity of Ioannina, Greece
http://dmod.cs.uoi.gr
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Why diversify?
Car
Animal
Sports Team“Mr.
Jaguar’’
DMOD lab, University of Ioannina 3[1] Marina Drosou, Evaggelia Pitoura: Search result diversification. SIGMOD Record 39(1): 41-
47 (2010)
What it means Given a set P of query results we want to select a
representative diverse subset S of P
What diverse means[1]? Coverage: different aspects, perspectives,
concepts as in the example of web search
Dissimilarity: non-similar items e.g., a number of characteristics in recommendations
Novelty: items not seen in the past
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Shortcomings of previous approaches
where1. P = {p1, …, pn}2. k ≤ n3. d: a distance metric4. f: a diversity
function
),(argmax* dSfS k|S|
PS
Given a set P of items and a number k, select a subset S* of P with the k most diverse items of
P.
Most previous work views as a top-k problem
Find:
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What is the right size for the diverse subset S? What is a good k?
What if… instead of k, a radius r?
Given a result set P and a radius r, we select a representative subset S ⊆ P such that:1. For each item in P, there is at least one similar
item in S (coverage)2. No two items in S are similar with each other
(dissimilarity)
Our approach - DisC Diversity
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r-DisC set: r-Dissimilar and Covering set
Zoom-out
Zoom-in Local zoom
Small r: more and less dissimilar points (zoom in) Large r: less and more dissimilar points (zoom out) Local zooming at specific points by adjusting the radius around them
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Talk Overview
Formal definition and algorithms
Comparison
Adaptive Diversification
Implementation using M-trees
Evaluation
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Our approach - DisC Diversity
Since a DisC set for a set P is not unique We seek a concise representation → the minimum DisC set
Let P be a set of objects and r, r ≥ 0, a real number. A subset S ⊆ P is an r-Dissimilar-and-Covering diverse subset, or r-DisC diverse subset, of P, if the following two conditions hold:1. (coverage condition) ∀pi ∈ P, ∃pj ∈ N+
r (pi), such that pj ∈ S and
2. (dissimilarity condition) ∀ pi, pj ∈ S with pi ≠ pj , it holds that d(pi, pj) > r
Formal definition:
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Graph model We use a graph to model the problem:
Each item is a vertex There exists an edge between two vertices, if
their distance is less than r
r
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Graph model Solving the minimum r-DISC DIVERSE SUBSET
PROBLEM for a set P is equivalent to finding a minimum Independent Dominating set of the graph. Independent: no edge between any two vertices in the set Dominating: all vertices outside connected with at least
one inside NP-hard
Dominating, not independent
Dominating and independent
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Computing DisC subsets
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How smaller is the minimum set?
where B the maximum number of independent neighbors of any item in P
i.e., each item has at most B neighbors that are independent from each other.
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The size of any r-DisC diverse subset S of P is B times the size of any minimum r-DisC diverse subset S∗
B depends on the distance metric and data cardinality
We have proved that: for the Euclidean distance in the 2D plane: B = 5 for the Manhattan distance in the 2D plane: B = 7 for the Euclidean distance in the 3D plane: B = 24
(proofs in the paper)
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Bounding the size of DisC subsets
Raising the dissimilarity condition:
Let Δ be the maximum number of neighbors of any item in P. The size of any covering (but not dissimilar) diverse subset S of P is at most lnΔ times larger than any minimum covering subset S∗
(proof in the paper)
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Talk Overview
Formal definition and algorithms
Comparison
Adaptive Diversification
Implementation using M-trees
Evaluation
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Comparison with other models
Two widespread options for f:
),(min),( ,MIN ji
ppSpp
ppddSfji
ji
Spp
jiji
ppddSf ,
SUM ),(),(
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Comparison with other models
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Comparison with other models
Let S be an r-DisC set and S* be an optimal MAXMIN set. Let and * be the MAXMIN distances of the two sets. Then, * ≤ 3.
(proof in the paper)
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Talk Overview
Formal definition and Algorithms
Comparison
Adaptive Diversification
Implementation using M-trees
Evaluation
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Zooming We want to change the radius r to r’ interactively and
compute a new diverse set r’ < r zoom in, r’ > r, zoom out
Two requirements:1. Support an incremental mode of operation:
the new set Sr’ should be as close as possible to the already seen result Sr. Ideally, Sr’ ⊇ Sr for r’ < r and Sr’ ⊆ Sr for r’ > r
2. The size of Sr’ should be as close as possible to the size of the minimum r’-DisC diverse subset
There is no monotonic property among the r-DisC diverse and the r’-DisC diverse subsets of a set of objects P (the two sets may be completely different)
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Size when moving from r -> r’
The change in size of the diverse set when moving from r to r’ depends on the number of independent neighbors (for r’) in the “ring” around an object between the two radii.
𝑁 (𝑝𝑖)𝑟 1 ,𝑟 2
𝐼
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Zooming
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Again, depends on the distance metric and data cardinality
2D Euclidean
2D Manhattan
(proofs in the paper)
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Zooming-In For zooming-in, we keep the items of Sr and fill in the solution
with items from uncovered areas.
It holds that:1. Sr ⊆ Sr′
2. |Sr′| ≤ N|Sr|, where N is the maximum in Sr
(proofs and algorithms in the paper)
(proof and various algorithms for keeping the size small in the paper)
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Zooming-Out For zooming-out, we keep the independent items
of Sr and fill in the solution with items from uncovered areas.
It holds that:1. There are at most N items in Sr\Sr’
2. For each item in Sr\Sr’, at most (B-1) items are added to Sr’
(proof and various algorithms for keeping the size small in the paper)
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Talk Overview
Formal definition and Algorithms
Comparison
Adaptive Diversification
Implementation using M-trees
Evaluation
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Implementation We base our implementation on a spatial data
structure (central operation: compute neighbors)
We use an M-tree We link together all leaf nodes (we visit items in a single
left-to-right traversal of the leaf level to exploit locality) We build trees using splitting policies that minimize
overlap
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Implementation
Lazy variations for updating neigborhoods
Our code is available on-line: www.dbxr.org (VLDB 2013 Reproducible label)
Pruning Rule: A leaf node that contains no white objects is colored grey. When all its children become grey, an internal node is colored grey and becomes inactive. We prune subtrees with only “grey nodes”.
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PerformanceMany real and synthetic datasets
General trade-off:Larger r → Smaller diverse set → higher cost
Lazy variations of our algorithms further reduce computational cost
The cost also depends on the characteristics of the M-tree (fat-factor)
Smaller sizes for clustered data
Cost
Solution size
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Zooming performanceSolution size
Cost
Jaccard distance among solutions
Both requirements: incremental (much smaller cost) and small size (relative to computing it from scratch)
Larger overlap among Sr and Sr’
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On-going and future work
1. Incorporate relevance: instead of locating the smaller set, locating the
“most relevant” set
2. Use multiple radii: emphasize specific areas of the dataset emphasize specific items, e.g., most relevant
3. Streaming (publish/subscribe) systems: also “novelty”
Many other – other forms of indexing, integrating the notion of diversity with database query processing, etc .
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Thank you!
See DisC and other models in action in our demo! Poikilo @ Group D
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Computing DisC subsets Let us call black the objects of P that are in S,
grey the objects covered by S and white the objects that are neither black nor grey.
Initially, S is empty and all objects are white. ▫until there are no more white objects.
select an arbitrary white object pi
color pi black and colors all objects in the neighborhood of pi grey.
Greedy variation:▫At each step, we select the white object with the
largest number of white neighbors.